The present invention is directed to electrical computers and data processing systems, with applications involving finance. More particularly, the present invention includes an apparatus, along with methods for making and using it, to receive input data (which can represent market data), to process the input data to calculate projected data, and to generate output including the projected data, wherein the processing was first tested by calculating projected test data from input test data and then using the projected test data to derive a portion of the input test data. The projected data can include financial simulations of future market behavior, such as prices, to aid in making financial decisions including transactions, hedging, etc.
A. Overview
Many mathematical and statistical techniques have been used to estimate the likelihood of future events. Sophisticated techniques include xe2x80x9cRandom Walkxe2x80x9d models which assume that future behavior characteristics will continue as they have in the past. Thus, projections and distributions of projections can be made and, by examining historic data, a statistical level of confidence for the projections can be computed. Such models, which are usually implemented by computer, have been used as the basis for making financial decisions.
To intelligently engage in market transactionsxe2x80x94buying or selling a financial product, and even maintaining an investment positionxe2x80x94players considering possible future market behavior make projections from present market phenomenon. One aspect of such projections involves market simulations, wherein the future market prices of such financial products as futures, swaps, options, and any other derivative products are randomly generated in great numbers and over chosen future time periods.
A xe2x80x9cforward pricexe2x80x9d is a risk-adjusted future spot price; a xe2x80x9cfuture spot pricexe2x80x9d is the spot price to be observed at some future time; and the xe2x80x9cspot pricexe2x80x9d is the price for which some asset can be exchanged for money. In case of commodity markets, the xe2x80x98assetxe2x80x99 is some commodity; in case of equity markets, the asset is some stock; in case of interest rate markets, the xe2x80x98assetxe2x80x99 is some type of a loan or deposit. A financial xe2x80x9cderivativexe2x80x9d is a financial product having future cash flows, the values of which are derived as functions of future spot prices.
Financial products that commonly use simulation include: for Interest Rate Marketsxe2x80x94mortgage-rate contingent derivative products (e.g., derivative products for which future cash flows are derived as functions of future mortgages rates as xe2x80x98spot pricesxe2x80x99); path-dependent options, swaps, and swaptions (these are derivative products having future cash flows derived as functions of future London Inter Bank Offering Rates as xe2x80x98spot pricesxe2x80x99; and, counter-party risk exposure calculations (here, the xe2x80x98spot pricesxe2x80x99 can be any of the foregoing, but are combined with the additional default information of the counterparty); for Commoditiesxe2x80x94path-dependent options (which are derivative products having cash flows derived as functions of more than one future spot price, and where the future spot price is the price of the commodity at the corresponding future date), swaps and swaptions (these are derivative products having cash flows derived as functions of future spot prices, the spot prices being the commodity prices); and, counter-party-risk exposure calculations (which are functions of commodity spot prices and counterparty risks, but applied to commodity-related products); and for Equitiesxe2x80x94hedging scenarios (these are cash flows which result from using a particular market hedging strategy, with the cash flows derived as functions specific to the hedging strategy and of future equity prices as the xe2x80x98spot pricesxe2x80x99), and counter-party-risk exposures.
Simulations can also involve using present information about liquid financial products to predict forward prices and to generate price distributions of various liquid and illiquid financial products. As the number of random numbers increases, the average of simulated changes in prices xe2x80x9cconvergesxe2x80x9d toward the drift term, which is defined below.
Prices are typically considered as following xe2x80x9cBrownian motion.xe2x80x9d According to Brownian motion, a percent change in price depends on a deterministic drift term (i.e., an expected change in price) and a random term (which gives variability to price changes around the expected change in price). The future value of the random term cannot be predicted per se. However, the magnitude of price changes can be measured statistically as the price volatility over some previous time period. Thus, in simulating future prices, many random terms can be generated to build a future price distribution. The greater the number of random numbers generated, the closer the average of the simulated prices represents the drift term and the closer the simulated distribution represents the price volatility about the drift term.
At any point in time, the markets will provide quotes on the spot price and on a series of forward spot pricesxe2x80x94the quotes corresponding to a number of different future time periods. A market quote for a forward spot price corresponding to some future time period represents the market""s expectation of what the spot price will be at that future time periodxe2x80x94adjusted for risk.
The market quotes for the spot price and the forward prices combine to create what is called the xe2x80x9cforward price curve.xe2x80x9d Very seldom is the forward price curve the same from day to day, and it is the movement of the forward price curve which the simulations attempt to realistically represent.
Simulating methodologies typically use statistical parametersxe2x80x94such as price volatility and other characteristics of pride probability distributions to predict the distributions of various financial derivative product prices. Simulations of the distributions of derivative product future cash-flows can be used to solve a, variety of problems including pricing, hedging analysis, and profit-loss analysis, as set forth below.
1. Pricing
Simulations can be used in pricing financial products, even those which are difficult to price because they are illiquid (i.e., rarely traded) and thus do not have a readily available market price. Such financial products are difficult to price easily or correctly with readily available, simpler pricing techniques.
2. Hedging
Simulations can be a powerful financial tool for hedging analysis or portfolio management. The simulation of the market behavior allows for an analysis of particular hedging scenarios which a firm might consider for managing its exposure to market risks. In comparing different hedging scenarios, one would analyze the standard deviations of the simulated distributions: the smaller the standard deviation, the better is the hedging scenario.
3. Profit/Loss Analysis
Simulations can also be used to simulate the profit and loss distributions of portfolios of derivative products. The generated distribution of the portfolio performance may be used very generally to manage a firm""s exposure to the market risks. By incorporating the market risks with the counter-party default risks, simulations can be used to manage the firm""s exposure to the counter-party risks. Particular measures of this counter-party risk exposure can be used by the firm to make decisions on when to limit the firm""s dealing with some counter-party. For these purposes, one would analyze the xe2x80x9ctailsxe2x80x9d of the distribution curves which would represent unlikely but extreme events.
B. Methods of the Prior Art
Simulation methods are widely used in fields such as physics and finance. Through a method commonly referred to as xe2x80x9cMonte Carlo,xe2x80x9d a large number of random numbers are generated in simulating random behaviors. All Monte Carlo methods have in common an assumption that random behaviors can be represented by using a Random Walk model. To select a particular Random Walk model, performance of the model is tested by calculating confidence levels from historic data.
Specifically, in finance, Monte Carlo methods have been used to calculate expected prices for financial products. In general, pricing methods use such statistics to simulate forward price or forward cash flow distributions; these methods use the average of these distributions to predict the expected forward price or expected forward value of the cash flow.
Consider, for example, a financial product which has an uncertain future cash flow occurring at some known future time. The distribution that would be used to price this financial product would be a probability distribution of this future cash flow. Then, the average price of the distribution is the expected forward price of the financial product. The present value of this forward price would represent the price one would pay today in order to receive the uncertain future cash flow.
Methods of the prior art, which are almost always computerized, include a single-factor model, a two-factor model, and a multi-factor model. The most frequently used model is the single-factor model, followed closely by the two-factor model. While considered superior in theory, multi-factor models have not been in common use due to problems in applying them.
1. Single-factor Model
The simplest existing market simulation methods include the single-factor model. Typically, this model assumes that the historical distribution of spot prices provides all the information needed to determine the distribution of spot prices in the future. The single-factor in this model stands for a single distribution (of the spot price) being generated at every point in time.
The single-factor model is extremely simple in design and cannot incorporate present market information about future events. The simplicity of this model has to do with the fact that it assumes that there is a single variable (termed the xe2x80x9cdriver factorxe2x80x9d or the xe2x80x9cindependent variablexe2x80x9d) that moves around and cannot be exactly predicted. This single variable is assumed to drive all other prices (termed the xe2x80x9cdependent variablesxe2x80x9d).
An example of a single-factor model would be one having all the forward prices for a forward price curve move Unequally or in proportion to each other over time. This means that if the spot price goes up, all the forward pricesxe2x80x94the dependent variablesxe2x80x94on the curve at that particular time go up. See FIG. 1.
Typically, the driver factor is the spot price. Thus, the distribution of the spot price at some time in the future is simulated such that it is centered around today""s spot market price. In the case of the crude oil market, for example, the expected forward spot price of West Texas Intermediate (WTI) crude oil (in present dollar terms) would be, according to this single-factor model, today""s spot market price.
2. Two-factor Model
A two-factor simulation model can represent market behavior better by adding a second driver factor to drive the forward price curve. The important distinction here is that two things are allowed to be random, thus allowing a better representation of future market behavior. See FIG. 2.
An example of a two-factor model would be the case where the spot price is the first driver factor and some long-term forward price is taken to be the second driver factor. Now the curve could become steeper or flatter while at the same time the overall forward price level could go up or down.
3. Multi-factor Model
Finally, a multi-factor simulation model for the market prices brings variability into the whole curve of forward prices for any future calendar time (i.e., simulation node), as illustrated in FIG. 3.
The market provides quotes for current forward prices for different periods, thereby defining a curve for the particular market in question. This curve is used as the starting point for the multi-factor simulation. For the sake of the example, consider that the market has provided forward spot prices for ten different future dates of some particular interest rate, commodity, or equity; these are used to construct the forward price curve. Then, at the first future calendar period of interest (i.e., also known as the first xe2x80x9csimulation nodexe2x80x9d) ten forward price distributions can be randomly generated around the present market forward price values. Thus, the distribution of each forward price at some future time is characterized by its average, which is also the current market value of the forward price. Historical correlations between the movements of different futures prices along the curve can also be built into the multi-factor model.
The state of the art for known multi-factor models typically assumes that the distributions of the forward prices at any point in time are centered around the existing forward prices implied by today""s market. For example, if the current forward rate curve as quoted in the market indicates that the market forward rate for a three month loan effective one year from today (expressed in terms of general language used here, this forward rate would be the xe2x80x98forward spot pricexe2x80x99 for a future time which is one year away from today) is 5%, then the simulation will center all the three month rates with forward start times one year away at any simulation node at 5%. Typically, the variability of the forward prices is assumed constant, and is either given the current market option volatility quote value or a long-term historically calculated value.
C. Drawbacks with these Methods
Unfortunately, the above-described methods have drawbacks that have not been solved in the prior art. The primary test of the accuracy of a simulation model is how closely its answers correspond with market events, and these three existing methods fail this test due to errors in their basic assumptions.
1. Drawbacks of the Single-factor Model
A single-factor model, while easy to use due to its simplicity, is overly simple as it assumes that all market prices are driven by a single random independent variable. In reality, many market variables exhibit randomness, and market prices of financial derivative products are often functions of several if not all of these independent variables. Needless to say, the simulations of markets through the use of a single-factor model give prices of financial derivative productsxe2x80x94as the dependent variablesxe2x80x94which do not, when tested, all simultaneously converge to their present market prices. (Convergence should occur as the number of random values generated and used in the creation of the price distributions increases.) Accordingly, applying the same single-factor simulations to different financial products would put the user at the risk of arbitrage: using these simulations to price a variety of derivative products would put the user at financial risk from dealing with somebody whose more realistic simulations converge just slightly closer to the market values. The analogy could be made to an xe2x80x98insider/outsiderxe2x80x99 traderxe2x80x94one who uses simulations which reflect a better knowledge of market behavior versus a trader with a simulation model that reflects very little about the actual market behavior. Accordingly, using the single-factor model as a guide in making financial transactions, and even in making the decision to maintain an investment position, would not be as financially productive as would using a more accurate model.
In summary, the problems with the single-factor model are as follows: (i) simulations do not converge to market prices; (ii) a single driver-factor is inadequate to represent the complicated market reality; (iii) the model does not provide a means for correctly simulating and pricing illiquid products as dependent variables using liquid product data; and (iv) financial transaction decisions based on the model are not as good as they would be if a more accurate model were used.
2. Drawbacks of the Two-factor Model
While the two-factor model captures much more of the market reality through the additional driver factor, it still overly simplifies market behavior. It also does not provide for simultaneous convergence to market prices of a variety of financial products as dependent variables as two factors is still not enough to fully explain the manner in which the forward price curves move over time. In other words, the market forward price curves are in reality driven by much more than just two driver factors.
In summary, the problems with the two-factor model are the same as-with the single-factor model, those being: (i) averages of distributions of do not converge to market prices; (ii) while there is an improvement over a single-factor, two factors are still inadequate to represent the complicated market reality; and (iii) the model does not provide a means to simulate the behavior of and price illiquid products using liquid product data; and (iv) financial transaction decisions based on the model are not as good as they would be if a more accurate model were used.
3. Drawbacks of Multi-factor Model
Multi-factor models have the most potential for representing the complex behavior of the market because they do not limit the number of driver factors. These models typically use ten to twenty such driver factors.
Multi-factor models attempt to address two of the problems with the single- and two-factor models. With regard to the problem of simulating illiquid product prices using liquid product data, multi-factor models have the theoretical ability to break market behavior down into driver factors that can then be used to define the relationships between liquid and illiquid productsxe2x80x94relationships that in turn can be used for simulating the behavior of any financial productxe2x80x94liquid and/or illiquid. To date, however, no model is known to have been able to actually perform this task.
Also, the fundamental problem of market convergence, continues to be an issue. Existing methods have not provided accurate price predictions for liquid products. By extension, if these models cannot predict accurate liquid product prices as represented by market prices, they can not accurately predict illiquid product prices using liquid data. In order to use liquid market data to simulate the behavior of illiquid product prices, the convergence of the method back to observed liquid product market prices is used.
A primary cause for the lack of market convergence is that, generally, the multi-factor models center all forward spot price distributions around the current spot market price. This assumption is contrary to the way in which market prices are determined. In reality, the current market forward prices represent the risk-adjusted future spot prices. This information is ignored by current technology. Risk-adjusted future spot prices need to be taken into account appropriately by the simulation technology, and existing multi-factor models fail to do so.
Furthermore, there is the issue of the volatility of the forward spot prices that are used as the driver factors and, in particular, how that volatility changes over time. The current multi-factor simulation technology does not address this issue and generally assumes that the variability of each forward spot price is constant over time. In reality, however, the market prices of derivative productsxe2x80x94the dependent variablesxe2x80x94tend to reflect the changing volatilities of the driver factors.
Unless the volatilities of forward prices as the independent variables are allowed to change over time, the simulated behaviors of dependent variables turn out to be inaccurate. In fact, holding forward price volatilities constant (as most models do) has the following negative effects: (i) this is not what is observed in historical dataxe2x80x94thus the assumption of constant volatilities contradicts the statistically observed volatility behavior; (ii) this is not what is observed and implied by market prices of optionsxe2x80x94thus the assumption of constant volatilities contradicts the market expectations of how that volatility will behave over some future time period; and (iii) the bridge between the liquid option volatility market information and the historical forward price volatilities cannot be built without allowing the volatilities to change over timexe2x80x94as constant volatilities contradict both the historical and the current market observations and thus also ignore any meaningful relationship possibilities between the two observations. Thus, as with other models of the known prior art, financial decisions based on the use of a multi-factor model are not as good as they would be if they were based on a more accurate model.
The problems with existing multi-factor models can therefore be summarized as follows: (i) the averages of distributions of dependent variables do not converge to market prices; and (ii) while existing multi-factor models provide a means for simulating and pricing illiquid products-sometimes using liquid product data, these models fail to simulate liquid products in convergence back to the same market data used as inputs, which guarantees inaccuracies in simulating behaviors of other liquid or illiquid pricesxe2x80x94if the simplest liquid products cannot be priced correctly, then correct pricing of more complicated liquid or illiquid products cannot be expected; (iii) the models exclusively rely on current spot prices for developing future spot price distributions, thus ignoring the market information provided by the market forward prices which include information about future spot prices; (iv) existing methods are typically unable to account for changing volatilities of driver factors and thereby make the inaccurate assumption that these driver factor volatilities remain constant over time; and (v) financial transaction decisions based on the model are not as good as they would be if a more accurate model were used.
In sum, then, the above-referenced prior art has not yet uncovered a system for simulating, pricing, and hedging financial products without significant drawbacks and limitations.
A. Objects of the Invention
In general, the present invention is intended to have utility in addressing problems, particularly those inherent in the existing multi-factor simulation methods, and as an improvement over the prior technology, as indicated by the following additional, representative objects of the invention.
An object of the present invention is to provide a computerized system for simulating, pricing, and hedging financial products.
It is also an object of the present invention to provide a multi-factor computerized system for simulating, pricing, and hedging financial products with convergence to market price.
An additional object of the present invention is to provide a multi-factor computerized system for simulating, pricing, and hedging financial products in a manner consistent with data inputs.
A further object of the present invention is to provide a multi-factor computerized system for simulating, pricing, and hedging financial products with consistency among simulations of driver factor behavior and dependent variable behavior.
Another object of the present invention is to provide a multi-factor computerized system for simulating behavior of, generating distribution of, pricing, and hedging financial products, the system having consistency among final outputs.
Yet another object of the present invention is to provide a model that uses market information provided by the market forward prices, which includes information about future spot prices.
Still another object of the present invention is to provide a model that accounts for changing volatilities of driver factors.
Still another object of the present invention is to provide a model that facilitates financial transaction decisions.
Other objects and advantages of the present invention will become apparent from the following summary of the invention, drawings, and detailed description of a preferred embodiment of the present invention.
B. SUMMARY OF THE INVENTION
In accordance with the objects of the present invention, a computer system for generating and testing projected data is provided. The system includes a digital computer connected to means for receiving input data for making projections about a first variable and means for outputting processed data; and logic means for controlling the digital computer. The logic means implements a mathematical technique underlying the present invention. The logic means uses the technique to process the input data to calculate projected data, to test the accuracy of the projected data by calculating the input data from the projected data, and to generate output including the tested projected data. The projected data can include volatilities for the projected data.
The logic means can also use the technique to generate distributions of the variable. The average of each variable distribution converges to the projected data as the number of simulated projected data generated to form the distribution increases. And the simulation also can have distributions of the volatilities, each having an average. The simulations are generated so that the averages converge to the volatilities with an increase in the number of volatilities generated.
The system can be applied to simulating behavior of, generating distribution of, pricing, and hedging financial products, and for supporting financial decisions, such as whether to make a particular transaction. The system, termed for recognition in the market the xe2x80x9cUnivol System,xe2x80x9d is a multi-factor model which uses forward prices as the driver factors. However, the Univol System improves over the prior art as set forth below.
1. Convergence with Market Price
No known existing technology converges the averages of generated distributions to the input data, e.g., converges output expected prices to prices actually observed in the market. However, the present invention starts with the current market prices, uses them as inputs to the system, and then works backwards to build relationships between observed and simulated variables, thereby converging the expected prices on the market prices.
2. Consistency Between Data Inputs, Intermediary Simulations, and Final Outputs
The present invention uses data. as inputs to the process of defining driver factors. The definitions are then tested for accuracy by calculating the input data from the projected data. When used in a financial application, say, to facilitate a decision to buy, sell, or keep a financial product, the input data includes the most recent liquid market data. The computer system is then used to compute present market values and their distributions, which are then used as a basis for making a buy, sell, keep decision.
Because the averages of distributions of financial products, the market prices of which were used as input data, converge back to the input market prices, there is consistency between the market data that is input and the simulated market behavior of the liquid products.
The driver factors, which can be built using the information from the liquid market data, are then used to simulate the illiquid market data. Therefore, both the liquid and illiquid products in a sense have a common denominatorxe2x80x94the driver factors. Thus, there is consistency between the treatment of liquid and illiquid products.
3. More Realistic Price Distributions
Prior art methods generally center all forward spot price distributions around a single price, namely a particular day spot price. In the present invention, however, the simulated forward spot prices for any time T in the future are driven by the market expectation of future market spot prices for the same time Txe2x80x94risk adjusted. This distribution-centering strategy more realistically reflects market expectations. By building distributions around these market expectations, the expected variability of markets is better incorporated.
The present invention recognizes that the expected spot price at some time T in the futurexe2x80x94risk-adjustedxe2x80x94is today""s forward price expiring at time T. Thus, the simulated spot price distribution for future calendar time T is centered around today""s forward price expiring at time T.
One can visualize this process as follows: the expected forward price curve retains its shape but moves to the left over future calendar time. (See FIGS. 4a-4c.) For example, today the spot price is quoted at $95 and the three-month forward price is quoted at $100. Then, three months from now, the present invention would project the risk-adjusted spot price to be $100, whereas other methods might project the value to be $95.
4. Allows Volatilities to Change
The present invention makes use of a tendency of the current period forward price volatilities to revert back to the long-term historical volatilities. Thus, the present invention is able to capture changes in volatility values over time by dynamically relating the current period forward price volatilities to these long-term historical volatilities.
5. Supports Financial Decisions
The present invention facilitates those financial decisions that have been made using the models of the above-mentioned prior art, except that the decisions are made more accurately. This accuracy should be reflected in increased profits from transactions made (and refrained from) based upon the information generated by the present invention.
Indeed, a wide spread use of this invention ought to make the markets more efficient and the bid/ask spreads on swaptions (an example of an illiquid financial product) tighter, as the invention would provide a means for pricing the swaptions in a manner that is consistent with the pricing of all other liquid derivative productsxe2x80x94and thus would eliminate the guess-work which is currently involved in the pricing of illiquid products, such as swaptions.